Properties

Label 980.1.ba.b
Level $980$
Weight $1$
Character orbit 980.ba
Analytic conductor $0.489$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [980,1,Mod(239,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 7, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.239");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.ba (of order \(14\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.110730297608000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{14} q^{2} + ( - \zeta_{14}^{6} - 1) q^{3} + \zeta_{14}^{2} q^{4} - \zeta_{14}^{3} q^{5} + ( - \zeta_{14} + 1) q^{6} - \zeta_{14}^{2} q^{7} + \zeta_{14}^{3} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{5} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{14} q^{2} + ( - \zeta_{14}^{6} - 1) q^{3} + \zeta_{14}^{2} q^{4} - \zeta_{14}^{3} q^{5} + ( - \zeta_{14} + 1) q^{6} - \zeta_{14}^{2} q^{7} + \zeta_{14}^{3} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{5} + 1) q^{9} - \zeta_{14}^{4} q^{10} + ( - \zeta_{14}^{2} + \zeta_{14}) q^{12} - \zeta_{14}^{3} q^{14} + (\zeta_{14}^{3} - \zeta_{14}^{2}) q^{15} + \zeta_{14}^{4} q^{16} + ( - \zeta_{14}^{6} + \zeta_{14} - 1) q^{18} - \zeta_{14}^{5} q^{20} + (\zeta_{14}^{2} - \zeta_{14}) q^{21} + ( - \zeta_{14}^{6} + \zeta_{14}^{5}) q^{23} + ( - \zeta_{14}^{3} + \zeta_{14}^{2}) q^{24} + \zeta_{14}^{6} q^{25} + ( - \zeta_{14}^{6} - \zeta_{14}^{5} - \zeta_{14}^{4} - 1) q^{27} - \zeta_{14}^{4} q^{28} + ( - \zeta_{14}^{3} + 1) q^{29} + (\zeta_{14}^{4} - \zeta_{14}^{3}) q^{30} + \zeta_{14}^{5} q^{32} + \zeta_{14}^{5} q^{35} + (\zeta_{14}^{2} - \zeta_{14} + 1) q^{36} - \zeta_{14}^{6} q^{40} + (\zeta_{14}^{4} + \zeta_{14}^{2}) q^{41} + (\zeta_{14}^{3} - \zeta_{14}^{2}) q^{42} + (\zeta_{14}^{5} + \zeta_{14}^{3}) q^{43} + ( - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14}) q^{45} + (\zeta_{14}^{6} + 1) q^{46} + (\zeta_{14}^{5} - \zeta_{14}^{4}) q^{47} + ( - \zeta_{14}^{4} + \zeta_{14}^{3}) q^{48} + \zeta_{14}^{4} q^{49} - q^{50} + (\zeta_{14}^{6} - \zeta_{14}^{5} - \zeta_{14} + 1) q^{54} - \zeta_{14}^{5} q^{56} + ( - \zeta_{14}^{4} + \zeta_{14}) q^{58} + (\zeta_{14}^{5} - \zeta_{14}^{4}) q^{60} + (\zeta_{14}^{2} - \zeta_{14}) q^{61} + ( - \zeta_{14}^{2} + \zeta_{14} - 1) q^{63} + \zeta_{14}^{6} q^{64} - q^{67} + (\zeta_{14}^{6} - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{69} + \zeta_{14}^{6} q^{70} + (\zeta_{14}^{3} - \zeta_{14}^{2} + \zeta_{14}) q^{72} + ( - \zeta_{14}^{6} + \zeta_{14}^{5}) q^{75} + q^{80} + (\zeta_{14}^{6} + \zeta_{14}^{5} - \zeta_{14}^{4} - \zeta_{14}^{3} + 1) q^{81} + (\zeta_{14}^{5} + \zeta_{14}^{3}) q^{82} + ( - \zeta_{14}^{4} + \zeta_{14}) q^{83} + (\zeta_{14}^{4} - \zeta_{14}^{3}) q^{84} + (\zeta_{14}^{6} + \zeta_{14}^{4}) q^{86} + ( - \zeta_{14}^{6} + \zeta_{14}^{3} - \zeta_{14}^{2} - 1) q^{87} + (\zeta_{14}^{4} - \zeta_{14}) q^{89} + ( - \zeta_{14}^{4} + \zeta_{14}^{3} - \zeta_{14}^{2}) q^{90} + (\zeta_{14} - 1) q^{92} + (\zeta_{14}^{6} - \zeta_{14}^{5}) q^{94} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{96} + \zeta_{14}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 5 q^{3} - q^{4} - q^{5} + 5 q^{6} + q^{7} + q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 5 q^{3} - q^{4} - q^{5} + 5 q^{6} + q^{7} + q^{8} + 4 q^{9} + q^{10} + 2 q^{12} - q^{14} + 2 q^{15} - q^{16} - 4 q^{18} - q^{20} - 2 q^{21} + 2 q^{23} - 2 q^{24} - q^{25} - 3 q^{27} + q^{28} + 5 q^{29} - 2 q^{30} + q^{32} + q^{35} + 4 q^{36} + q^{40} - 2 q^{41} + 2 q^{42} + 2 q^{43} - 3 q^{45} + 5 q^{46} + 2 q^{47} + 2 q^{48} - q^{49} - 6 q^{50} + 3 q^{54} - q^{56} + 2 q^{58} + 2 q^{60} - 2 q^{61} - 4 q^{63} - q^{64} - 12 q^{67} - 4 q^{69} - q^{70} + 3 q^{72} + 2 q^{75} + 6 q^{80} + 2 q^{81} + 2 q^{82} + 2 q^{83} - 2 q^{84} - 2 q^{86} - 3 q^{87} - 2 q^{89} + 3 q^{90} - 5 q^{92} - 2 q^{94} - 2 q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(\zeta_{14}^{6}\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.900969 + 0.433884i
0.900969 0.433884i
0.222521 0.974928i
−0.623490 0.781831i
−0.623490 + 0.781831i −1.62349 0.781831i −0.222521 0.974928i −0.900969 0.433884i 1.62349 0.781831i 0.222521 + 0.974928i 0.900969 + 0.433884i 1.40097 + 1.75676i 0.900969 0.433884i
379.1 0.222521 + 0.974928i −0.777479 0.974928i −0.900969 + 0.433884i 0.623490 + 0.781831i 0.777479 0.974928i 0.900969 0.433884i −0.623490 0.781831i −0.123490 + 0.541044i −0.623490 + 0.781831i
519.1 0.900969 + 0.433884i −0.0990311 0.433884i 0.623490 + 0.781831i −0.222521 0.974928i 0.0990311 0.433884i −0.623490 0.781831i 0.222521 + 0.974928i 0.722521 0.347948i 0.222521 0.974928i
659.1 0.900969 0.433884i −0.0990311 + 0.433884i 0.623490 0.781831i −0.222521 + 0.974928i 0.0990311 + 0.433884i −0.623490 + 0.781831i 0.222521 0.974928i 0.722521 + 0.347948i 0.222521 + 0.974928i
799.1 0.222521 0.974928i −0.777479 + 0.974928i −0.900969 0.433884i 0.623490 0.781831i 0.777479 + 0.974928i 0.900969 + 0.433884i −0.623490 + 0.781831i −0.123490 0.541044i −0.623490 0.781831i
939.1 −0.623490 0.781831i −1.62349 + 0.781831i −0.222521 + 0.974928i −0.900969 + 0.433884i 1.62349 + 0.781831i 0.222521 0.974928i 0.900969 0.433884i 1.40097 1.75676i 0.900969 + 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
49.e even 7 1 inner
980.ba odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.ba.b yes 6
4.b odd 2 1 980.1.ba.a 6
5.b even 2 1 980.1.ba.a 6
20.d odd 2 1 CM 980.1.ba.b yes 6
49.e even 7 1 inner 980.1.ba.b yes 6
196.k odd 14 1 980.1.ba.a 6
245.p even 14 1 980.1.ba.a 6
980.ba odd 14 1 inner 980.1.ba.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.ba.a 6 4.b odd 2 1
980.1.ba.a 6 5.b even 2 1
980.1.ba.a 6 196.k odd 14 1
980.1.ba.a 6 245.p even 14 1
980.1.ba.b yes 6 1.a even 1 1 trivial
980.1.ba.b yes 6 20.d odd 2 1 CM
980.1.ba.b yes 6 49.e even 7 1 inner
980.1.ba.b yes 6 980.ba odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 5T_{3}^{5} + 11T_{3}^{4} + 13T_{3}^{3} + 9T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{6} - 5 T^{5} + 11 T^{4} - 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} - 2 T^{5} + 4 T^{4} - T^{3} + 2 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( (T + 2)^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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